Why does that matter? There's no way to ensure your employees aren't writing down their passwords. If you don't trust someone to follow procedure then don't employ them.

Security always exists in layers. Policy is a layer - e.g. asking people to do something. Another layer is to have a system to remind/enforce that policy. Writing down passwords is (generally) an obviously stupid thing to do, perhaps worthy of termination. But forgetting to put a passphrase on one SSH key when you regularly use a dozen of them for different purposes - this is absolutely not an offense worthy of termination.

If that's what you need it's possible to generate and issue keys from a centralized host just like with TLS. You can even set up login with SSH/TLS certs, which can be password protected.